Articles of algebra precalculus

Prove by induction that $\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$

As the title says I need to prove the following by induction: $$\sum_{i=1}^{2^n} \frac{1}{i} \ge 1+\frac{n}{2}, \forall n \in \mathbb N$$ When trying to prove that P(n+1) is true if P(n) is, then I get that: $$\sum_{i=1}^{2^{n+1}} \frac{1}{i} = (\sum_{i=1}^{2^n} \frac{1}{i} ) +(\sum_{i=1}^{2^n} \frac{1}{2^n+i} ) \ge 1+\frac{n}{2} + (\sum_{i=1}^{2^n} \frac{1}{2^n+i} )$$ using the inductive hypothesis. […]

Why can't equations with unknown inside and outside of a function be solved in a standard way?

For example the equation $$ n2^n = 8 $$ Is true for $n=2$ which can be guessed, but more complicated examples would require some sort of approach. Also with trigonometric functions, $$ x\sin(x) + B\cos(x) = A $$ I read that solutions of these kinds of equations can not be expressed in closed form, why […]

Solving the equation $\ln(x)=-x$

I tried solving this equation for a long time but did not succeed. Any help is appreciated. $$\ln x=-x$$ I am not sure the tag is correct, I am not familiar with English mathematical terms. Please correct me if I am wrong.

Expressing the minimum function in terms of the absolute value in a symmetric manner (generalized to more variables)

It is well known that: $$\max(a,b) = \frac12(a+b)+\frac12|a-b|$$ and similarly: $$\min(a,b) = \frac12(a+b)-\frac12|a-b|.$$ In fact, they are equivalent since $\max(a,b) = -\min(-a,-b)$. We can try generalizing these representations to more arguments. One obvious approach is to do the following: $$\max(a,b,c) = \max(a,\frac12|b+c|+\frac12|b-c|)$$ and then again use the previous expression. However, the final expression is quite […]

Sequence $a_k=1-\frac{\lambda^2}{4a_{k-1}},\ k=2,3,\ldots,n$.

Consider the sequence $a_1, a_2,\ldots,a_n$ with $a_1=1$ and defined recursively by $$a_k=1-\frac{\lambda^2}{4a_{k-1}},\ k=2,3,\ldots,n.$$ Find $\lambda>1$ such that $a_n=0$. The answer is $\lambda=\dfrac1{\cos\frac{\pi}{n+1}}$.

How do I transform the equation based on this condition?

If a and b are the roots of the equation $$2x^2-px+7=0$$ Then a-b is a root of ?

Is the square root of $4$ only $+2$?

This question already has an answer here: Is it wrong to say $ \sqrt{x} \times \sqrt{x} =\pm x,\forall x \in \mathbb{R}$? 6 answers

Show that the equation $x^2+y^2+z^2= (x-y)(y-z)(z-x)$ has infinitely many solutions in integers $x, y, z$.

Show that the equation $x^2+y^2+z^2= (x-y)(y-z)(z-x)$ has infinitely many solutions in integers $x, y, z$. It seems like if I find a set of $x,y,z$ that satisfy this for any values that will prove it. But how do I find that set?

Find $\Big\{ (a,b)\ \Big|\ \big|a\big|+\big|b\big|\ge 2/\sqrt{3}\ \text{ and }\forall x \in\mathbb{R}\ \big|a\sin x + b\sin 2x\big|\le 1\Big\}$

Find all (real) numbers $a $ and $b$ such that $|a| + |b| \ge 2/\sqrt{3} $ and for any $x$ the inequality $|a\sin x + b \sin 2x | \le 1$ holds. In other words, find the set $Q$ defined as $$Q = \Big\{\ (a,b)\ \Big| \quad 1. \left|a\right| + \left|b\right| \ge \frac{2}{\sqrt{3}}, \ \text{ […]

Simplifying hyperbolic compositions like $\sinh (N \operatorname{acosh} a)$

In many occasions, we may meet hyperbolic functions, as well as their combined ones. I want to simplify expressions like $$ \tanh\left( N\left(\textrm{acosh}~ a\right)\right) $$ and $$ \sinh\left( N\left(\textrm{acosh}~ a\right)\right) $$ for any positive integer $N$. My WAY: By the substitution $b=\textrm{acosh}~ a$, then $a=\cosh b$, and we have $$ \tanh\left( N\left(\textrm{acosh}~ a\right)\right) = \tanh\left( […]